A -torus knot
is obtained by looping a string through the hole of a torus
times with
revolutions before joining its ends, where and
are relatively prime. A -torus knot is equivalent to a -torus knot. All torus knots are prime
(Hoste et al. 1998, Burde and Zieschang 2002). Torus knots are all chiral,
invertible, and have symmetry group (Schreier 1924, Hoste et al. 1998).
Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are torus knots using the function KnotData[knot,
"Torus"].
(Williams 1988, Murasugi and Przytycki 1989, Murasugi 1991, Hoste et al. 1998). The unknotting number of a -torus knot is
(2)
(Adams 1991).
The numbers of torus knots with crossings are 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2,
1, ... (OEIS A051764). Torus knots with fewer
than 11 crossings are summarized in the following table (Adams et al. 1991)
and the first few are illustrated above.
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1991.Adams, C. C. The
Knot Book: An Elementary Introduction to the Mathematical Theory of Knots.
New York: W. H. Freeman, 1994.Burde, G. and Zieschang, H. Knots,
2nd rev. ed. Berlin: de Gruyter, 2002.Gray, A. "Torus Knots."
§9.2 in Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, pp. 209-215, 1997.Hoste, J.; Thistlethwaite,
M.; and Weeks, J. "The First Knots." Math. Intell.20, 33-48,
Fall 1998.Kronheimer, F. B. and Mrowka, T. S. "Gauge
Theory for Embedded Surfaces I." Topology32, 773-826, 1993.Kronheimer,
F. B. and Mrowka, T. S. "Gauge Theory for Embedded Surfaces II."
Topology34, 37-97, 1995.Murasugi, K. "On the Braid
Index of Alternating Links." Trans. Amer. Math. Soc.326, 237-260,
1991.Murasugi, L. and Przytycki, J. "The Skein Polynomial of a
Planar Star Product of Two Links." Math. Proc. Cambridge Philos. Soc.106,
273-276, 1989.Rolfsen, D. Knots
and Links. Wilmington, DE: Publish or Perish Press, 1976.Schreier,
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in "The On-Line Encyclopedia of Integer Sequences."Steinhaus,
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